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        <p>控制一台机械臂从给定目标点1运动到给定目标点2，通常分为<strong>机械臂逆解计算与选取</strong>，<strong>机械臂运动规划</strong>，<strong>机械臂路径规划</strong>3个步骤。</p>
<h1 id="逆解"><a href="#逆解" class="headerlink" title="逆解"></a>逆解</h1><p>机械臂逆解是根据给定世界中的坐标点，求出机械臂各关节角度的过程。逆解的计算通常分为<strong>几何法</strong>和<strong>代数法</strong>。几何法需要通过分析空间立体几何求解出到达目标点的<strong>所有情况</strong>，这为求解带来了困难，因为个人无法保证自己是否穷尽了所有可能的情况。</p>
<h2 id="逆解求解"><a href="#逆解求解" class="headerlink" title="逆解求解"></a>逆解求解</h2><p>代数法则借助<strong>正运动学矩阵</strong>与<strong>逆运动学矩阵</strong>相等这一思想来求解。例如，以0关节坐标系为基坐标系，第5关节在该基坐标系中的位姿矩阵可表示为$T_0^{5}$ ，那么$T_0^5$有两种表示形式：<br>$$<br>T_0^5&#x3D;T_0^1 \times T_1^2\times T_2^3 \times T_3^4\times T_4^5<br>$$</p>
<p>$$<br>T_0^5&#x3D;T_0^6 \times T_6^5<br>$$</p>
<p>通过比对两种方法求解出的位姿矩阵，就可以得到联立方程组。这样的方法可以得到许多方程组，我们需要找到容易求解的方程组（有相同元素，可快速消除冗余项）以便自己求解。</p>
<p>当解出第一个角度值后，该角度可以作为已知量代入方程中逐个解出其他角度，由于后解出角度依赖先解出的角度，那么做逆解选取时，就需要注意他们的依赖关系。</p>
<h2 id="逆解选取"><a href="#逆解选取" class="headerlink" title="逆解选取"></a>逆解选取</h2><p>由依赖关系组成的逆解，可以由树形结构表示。在代码中可以用数组实现树形结构。基础思想为每个关节都有一个<strong>备选解</strong>数组，通过遍历所有可能组合，求其正向运动学位姿矩阵，若正向运动学位姿矩阵与指定位姿<strong>相同</strong>或<strong>相差不大</strong>，则为可行解。判断是否相同或相似，通常分为以下2个步骤，步骤中的”差值”可人为设定，这是一个可调阈值：</p>
<ul>
<li>旋转矩阵各元素<strong>符号相同</strong>，且差值不大。</li>
<li>偏移矩阵各元素<strong>符号相同</strong>，且差值不大。</li>
</ul>
<p>经过上述方法，可得到包含所有可能得可行解的矩阵，接下来需要做最优解选取。通常以移动距离最小为条件选取最优解。</p>
<h1 id="运动规划"><a href="#运动规划" class="headerlink" title="运动规划"></a>运动规划</h1><p>从点$A$运动到点$B$，以怎样的方式运行呢？用多长时间运行到点$B$？能设定到达点$B$的加速度吗，这样就可以知道它有多大力了。答案是，可以的。这就是本文所讲的运动规划要干的事情。</p>
<p>最近的项目用的是5次多项式法来解决上述问题。实际控制中需要注意以下几点：</p>
<ul>
<li>应对每个关节单独进行规划，初始位置，初始速度，初始加速度，终止位置，终止速度，终止加速度，还有最重要的，<strong>步长</strong>。在教科书或者视频中，五次多项式中自变量常常为$t$(时间)，即你期望多久之后达到目标点。但我更愿意称呼它为<strong>拟合步长</strong>或者<strong>拟合分辨率</strong>。这个参数意味着你想要把每个关节的运动分为多少段。而多长时间执行多少段，是可以人为设定的。</li>
<li>每个关节进行单独规划后，可以对每个关节设定不同的拟合步长，只要执行时让其同步运动到最终位置即可。</li>
</ul>
<h1 id="路径规划"><a href="#路径规划" class="headerlink" title="路径规划"></a>路径规划</h1><p>下次再更。</p>

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